ABCM Symposium Series in Mechatronics - Vol. 5 Section III – Emerging Technologies and AI Applications Copyright © 2012 by ABCM Page 592 A NEW BIOGEOGRAPHY-BASED OPTIMIZATION APPROACH BASED ON SHANNON-WIENER DIVERSITY INDEX TO PID TUNING IN MULTIVARIABLE SYSTEM Marsil de Athayde Costa e Silva, [email protected] Undergraduate course of Mechatronics Engineering Camila da Costa Silveira, [email protected] Leandro dos Santos Coelho, [email protected] Industrial and Systems Engineering Graduate Program, PPGEPS Pontifical Catholic University of Parana, PUCPR Imaculada Conceição, 1155, Zip code 80215-901, Curitiba, Parana, Brazil Abstract. Proportional-integral-derivative (PID) control is the most popular control architecture used in industrial problems. Many techniques have been proposed to tune the gains for the PID controller. Over the last few years, as an alternative to the conventional mathematical approaches, modern metaheuristics, such as evolutionary computation and swarm intelligence paradigms, have been given much attention by many researchers due to their ability to find good solutions in PID tuning. As a modern metaheuristic method, Biogeography-based optimization (BBO) is a generalization of biogeography to evolutionary algorithm inspired on the mathematical model of organism distribution in biological systems. BBO is an evolutionary process that achieves information sharing by biogeography-based migration operators. This paper proposes a modification for the BBO using a diversity index, called Shannon-wiener index (SW-BBO), for tune the gains of the PID controller in a multivariable system. Results show that the proposed SW-BBO approach is efficient to obtain high quality solutions in PID tuning. Keywords: PID control, biogeography-based optimization, Shannon-Wiener index, multivariable systems. 1. INTRODUCTION One of the most popular controllers in industrial processes is the proportional-integral-derivative (PID) controller. This control strategy offers a simple and effective solution for many real problems. About 90% of the control problems are solved by using some type of PID controller (Levine, 1996). After its creation, around 1910, and the Ziegler-Nichols tuning methods (Ziegler and Nichols, 1942) the popularity of this kind of controller has grown. This is mainly because PID controllers have structure simplicity and meaning of the corresponding three parameters, which can be easily understood by process operators. Moreover, PID controllers have the advantage of good stability and high reliability. The use of evolutionary algorithms to tune gains of PID controllers has demonstrated ability of finding a set of good solutions (Iruthayarajan and Baskar, 2009). The evolutionary computation paradigms such as genetic algorithm (Altinten et al., 2008), differential evolution (Lianghong et al., 2008), evolution strategies (Iruthayarajan and Baskar, 2010), and evolutionary programming (Jiang and Ma, 2006) are able to find a reasonable solutions for problems in which classical methods based on gradient information cannot be applied or do not show good performance. Examples in control systems are presented in Fleming and Purshouse (2002). A recent approach called Biogeography-based Optimization (BBO) has shown promising results in solving of complex optimization problems (Simon, 2008). Biogeography is the science that studies the distribution of species in an ecosystem and how species arise or become extinct. The main contribution of this paper is validate a new of BBO approach that uses a diversity measurement to increase the capability of scape from local optima. In this work, the classical BBO and the proposed BBO based on diversity measurement are used to find the gains of a multivariable PID controller for a 2x2 industrial-scale polymerization reactor. The remainder of this paper is organized as follows: in section 2 are presented the basic concepts of PID control. Section 3 describes the BBO algorithm and the proposed approach. The formulation of the problem is detailed in section 4. Sections 5 and 6 present the results and conclusion, respectively. 2. PID CONTROL FOR MULTIVARIABLE SYSTEMS Consider the multivariable system (Multiple Inputs Multiple Outputs, MIMO) shown in Figure 1, where R(t) is the set of reference signals, Y(t) is the set of outputs and U(t) is the set of control signals. The error, e(t), is the difference between the output and the input signals. The control signals are calculated based on the error. The standard PID controller is described by equation (1). ABCM Symposium Series in Mechatronics - Vol. 5 Section III – Emerging Technologies and AI Applications Copyright © 2012 by ABCM Page 593 Figure 1. MIMO system with the PID control. de t)( ut()()= Ket + K etdt() + K (1) p i ∫ d dt where Kp, Ki and Kd are the proportional, integral, and derivative gains of the PID, respectively, and t is the time. Also, the integral and the derivative gains can be written as a function of the proportional gain: Ki=Kp/ti and Kd=Kptd, where ti and td are the integral and derivative time. The Laplace transform can be applied to the controller to give the following transfer function: = +1 + G( s) K p 1 td s (2) ti s where G(s) is the transfer function of the controller and the error is the input and the output is the control signal. Nevertheless, the derivative term of the controller can amplify some noisy signal and also causes a sudden elevation of the control signal when the set point changes. So a filter is applied to the derivative term of the controller to avoid these problems, thus the transfer function of the controller becomes the following: 1 t s G( s )= K 1 + + d (3) p t s t s i d +1 N where N is the filtering constant, normally used as a number between 4 and 20. For an nxn multivariable system H(s), equation (4), the controller becomes an nxn matrix as given by (5). L h11 ()() s hn1 s H() s = MOM (4) L h1n () s hnn ( s) L g11()() s gn1 s G() s = MOM (5) L g1n () s gnn () s where t s 1 dij g( s )= K 1 + + (6) ij pij t s td s iij ij +1 N In order to measure the performance of the controller four main kinds of performance criteria are usually considered: the integrated squared error (ISE), the integrated absolute error (IAE), the integrated time-weighted absolute error (ITAE) and the integrated time-weighted squared error (ITSE). These criteria are defined by the following equations: ∞ =( 2 +2 +K + 2 ) ISE et1()()() et2 etdtn (7) ∫0 ABCM Symposium Series in Mechatronics - Vol. 5 Section III – Emerging Technologies and AI Applications Copyright © 2012 by ABCM Page 594 ∞ = ( + +K + ) IAE et1()()( et 2 etdtn ) (8) ∫0 ∞ = ( + +K + ) ITAE tet1()()() tet2 tetdtn (9) ∫0 ∞ = ( 2 +2 +K + 2 ) ITSE tet1 )( tet2 )( tetn )( dt (10) ∫0 where ei is the error of the i-th output related to the i-th input. 3. BIOGEOGRAPHY-BASED OPTIMIZATION The BBO algorithm, proposed by Simon (2008), uses the concepts and models of biogeography. Furthermore, BBO approaches have demonstrated ability to solve and good convergence properties on various benchmark functions and engineering optimization problems (Rarick et al., 2009; Kumar et al., 2009; Kundra et al., 2009; Simon et al., 2009; Bhattacharya and Chattopadhyay, 2010; Gong et al., 2010). These models of biogeography describe how species migrate from a habitat to another one and how species arise or become extinct. Each solution used in the algorithm is considered as a habitat and has a habitat suitability index (HSI) that measure the suitability of the habitat. This index is related to aspects as, for example, rainfall, fauna and flora diversities, topography, and environment temperature. These aspects are also called suitability index variables (SIV). A good habitat has a high HSI, while a poor habitat has a low HSI. This means that good habitats have more good aspects than the poor ones. Habitats with high HSI have a high immigration rate due to their good aspects, whereas poor habitats have a low immigration rate but a high emigration rate unlike good ones. The migration rates are direct related to the number of species in a habitat. So, a habitat with many species has a high emigration rate, because it is almost saturated, while habitats with few species have high immigration rate because do not have good conditions to live in. This migration process increases the diversity of the habitat and the miscegenation and contributes to the species information sharing and the mutation probability. Figure 2 represents emigration and immigration as a function of the number of species. In the Figure 2, I and E represent the maximum rates of immigration and emigration, respectively, and S denotes the number of species. Figure 2. Emigration and immigration rates. These concepts inspired the proposition of BBO. In the algorithm the solutions are treated as habitats and their good aspects are shared based on the migration rates. The basic algorithm of BBO is described in the following lines. Step 1: Initialize the parameters used in the algorithm: Smax maximum number of species, E emigration rate, I the immigration rate, and mmax the maximum mutation rate. Step 2: Calculate the probability for each value of the number of species as follows: = 1 Pj (11) S max = where j 1,..., Smax , and P is the probability for the j-th habitat. Step 3: Generate an initial random set of habitats according to the constraints of the problem. Step 4: Start the loop: (4.i) Generate the immigration and emigration rates: ABCM Symposium Series in Mechatronics - Vol. 5 Section III – Emerging Technologies and AI Applications Copyright © 2012 by ABCM Page 595 − λ = I(1 j ) j (12) S max µ = j j E (13) Smax λ µ where j and j are the immigration and the emigration rates for the j-th habitat. (4.ii) Calculate the derivative probability: • = −λ + µ + µ = Ps ( s s)PP s s+1 s + 1 s 0 • = −λ + µ + λ + µ ≤ < (14) Ps ()s sP s s−1 P s − 1s+ 1 P s + 1 1 s Smax • = −λ + µ + λ = Ps ()s sPP s s−1 s − 1 s Smax (4.